
theorem Th43:
  for A being category, a being Object of A, x being set
  holds x in (Concretized A)-carrier_of a iff
  ex b being Object of A, f being Morphism of b,a
  st <^b,a^> <> {} & x = [f,[b,a]]
proof
  let A be category, a be Object of A, x be set;
  set B = Concretized A;
  reconsider ac = a as Object of B by Def12;
A1: x in the_carrier_of ac iff
  x in Union disjoin the Arrows of A & ac = x`22 by Def12;
A2: dom the Arrows of A = [:the carrier of A, the carrier of A:]
  by PARTFUN1:def 2;
  hereby
    assume
A3: x in B-carrier_of a;
    then
A4: x`2 in dom the Arrows of A by A1,CARD_3:22;
A5: x`1 in (the Arrows of A).(x`2) by A1,A3,CARD_3:22;
A6: x = [x`1,x`2] by A1,A3,CARD_3:22;
    consider b,c being object such that
A7: b in the carrier of A and c in the carrier of A and
A8: x`2 = [b,c] by A4,ZFMISC_1:def 2;
    reconsider b as Object of A by A7;
    take b;
    reconsider f = x`1 as Morphism of b,a by A1,A3,A5,A6,A8,MCART_1:85;
    take f;
    thus <^b,a^> <> {} & x = [f,[b,a]] by A1,A3,A5,A6,A8,MCART_1:85;
  end;
  given b being Object of A, f being Morphism of b,a such that
A9: <^b,a^> <> {} and
A10: x = [f,[b,a]];
A11: x`1 = f by A10;
A12: x`2 = [b,a] by A10;
  [b,a] in dom the Arrows of A by A2,ZFMISC_1:def 2;
  hence thesis by A1,A9,A10,A11,A12,CARD_3:22,MCART_1:85;
end;
